NIMCET 2009 — Mathematics PYQ

1. Identify the Curves

• Circle: $x^2+y^2=4$ represents a circle centered at the origin $(0,0)$ with a radius $r=2$.

• Line: $y=2-x$ (or $x+y=2$) is a straight line passing through points $(2,0)$ and $(0,2)$.

2. Find the Points of Intersection

Substitute $y=2-x$ into the circle equation:

The intersection points occur at $x=0$ and $x=2$.

• At $x=0$, $y=2$.

• At $x=2$, $y=0$.

  The line connects the intercepts of the circle in the first quadrant.

3. Determine the Required Area

The area bounded by the circle and the line in the first quadrant is the segment of the circle. This can be calculated as:

4. Perform the Calculation

• Area of the Quadrant: Since the total area of the circle is $\pi r^2$, the area of one quadrant is:

  $$\frac{1}{4}\pi (2{)}^2=\frac{1}{4}\pi (4)=\pi$$

• Area of the Triangle: The triangle is formed by the origin $(0,0)$ and the intercepts $(2,0)$ and $(0,2)$:

  $$\frac{1}{2}\times \text{base}\times \text{height}=\frac{1}{2}\times 2\times 2=2$$

5. Final Result

Correct Option:

(b) $(\pi -2)$ sq units